Question

Each limit represents the derivative of some function \(f\) at some number \(a\). State such an \(f\) and \(a\) in each case.

\(\mathop {{\rm{lim}}}\limits_{\theta \to \pi /6} \frac{{{\rm{sin}}\theta - \frac{1}{2}}}{{\theta - \frac{\pi }{6}}}\)

Short answer

\(f\left( \theta \right) = {\rm{sin}}\theta \) and \(a = \frac{\pi }{6}\)

Step-by-step solution

Derivative formula

Rewrite the given expression in the form \(f'\left( a \right) = \mathop {{\rm{lim}}}\limits_{x \to a} \frac{{f\left( x \right) - f\left( a \right)}}{{x - a}}\).

\(\begin{aligned}\mathop {{\rm{lim}}}\limits_{\theta \to \pi /6} \frac{{{\rm{sin}}x - \frac{1}{2}}}{{\theta - \frac{\pi }{6}}} &= \mathop {{\rm{lim}}}\limits_{x \to \pi /6} \frac{{{\rm{sin}}x - {\rm{sin}}\left( {\frac{\pi }{6}} \right)}}{{\theta - \frac{\pi }{6}}}\\ &= \mathop {{\rm{lim}}}\limits_{x \to \pi /6} \frac{{f\left( x \right) - {\rm{f}}\left( {\frac{\pi }{6}} \right)}}{{x - \frac{\pi }{6}}}\\ &= f'\left( {\frac{\pi }{6}} \right)\end{aligned}\)

Compare the expression

By Compare the obtained expression with \(f'\left( a \right) = \mathop {{\rm{lim}}}\limits_{x \to a} \frac{{f\left( x \right) - f\left( a \right)}}{{x - a}}\).

We observe that \(f\left( \theta \right) = {\rm{sin}}\theta \) and \(a = \frac{\pi }{6}\).

Thus, \(f\left( \theta \right) = {\rm{sin}}\theta \) and \(a = \frac{\pi }{6}\).